MATHEMATICAL ENGINEERING TECHNICAL REPORTS

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1 MATHEMATICAL ENGINEERING TECHNICAL REPORTS Truss Topology Optimization with a Limited Number of Different Cross-Sections: A Mixed-Integer Second-Order Cone Programming Approach Yoshihiro KANNO METR November 2014 DEPARTMENT OF MATHEMATICAL INFORMATICS GRADUATE SCHOOL OF INFORMATION SCIENCE AND TECHNOLOGY THE UNIVERSITY OF TOKYO BUNKYO-KU, TOKYO , JAPAN WWW page:

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3 Truss Topology Optimization with a Limited Number of Different Cross-Sections: A Mixed-Integer Second-Order Cone Programming Approach Yoshihiro Kanno Department of Mathematical Informatics, University of Tokyo, Tokyo , Japan Abstract In deign practice it is often that the structural components are selected from among easily available discrete candidates and a number of different candidates used in a structure is restricted to be small. Presented in this paper is a truss topology optimization method for finding the minimum compliance design in which only a limited number of different cross-section sizes are employed. The member cross-sectional areas are considered either discrete design variables that can take only predetermined values or continuous design variables. In both cases it is shown that the compliance minimization problem can be formulated as a mixed-integer secondorder cone programming problem. The global optimal solution of this optimization problem is then computed by using an existing solver based on a branch-and-cut algorithm. Numerical experiments are performed to show that the proposed approach is applicable to moderately large-scale problems. Keywords Topology optimization; global optimization; mixed-integer programming; second-order cone programming; optimal standardization. 1 Introduction This paper attempts to shed new light on a classical problem in the truss topology optimization. We consider the minimization problem of the compliance under the volume constraint. Within the framework of the ground structure approach, the member cross-sectional areas are considered the continuous design variables. Then we deal with an optimization problem for finding the stiffest truss design that consists of only a limited number of different member sizes (cross-sectional areas). It is known that the minimum compliance problem of trusses with continuous design variables can be recast as a second-order cone programming (SOCP) problem. This SOCP problem plays a fundamental role in formulating the optimization problems proposed in this paper. Based upon the minimum principle of complementary energy for the equilibrium analysis, the SOCP problem is formulated in terms of the member cross-sectional areas, axial forces, and the complementary strain energies. The second-order cone constraints stem from the definition of the complementary strain Address: Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo , Japan. kanno@mist.i.u-tokyo.ac.jp. Phone: Fax:

4 energy stored in each member. SOCP is a class of convex optimization and its optimal solution can be computed efficiently with a primal-dual interior-point method [2, 4]. Formulations based upon the minimum principle of complementary energy have sometimes been used in studies of compliance minimization problems of continua [3, 9, 21]. This type of formulation for trusses is due to Bendsøe et al. [8]. There, the problem was not recast as an SOCP problem. Moreover, one should use positive lower bounds for the member cross-sectional areas, because the complementary strain energy is inversely proportional to the member cross-sectional area. The SOCP formulation, however, does not require the positive lower bound; indeed, the cross-sectional areas are allowed to become equal to zeros in the optimization process. This SOCP formulation, briefly recalled in section 2.2 of this paper, can be found in Makrodimopoulos et al. [26]. The dual problem of this SOCP problem may be regarded as a type of minimum compliance problems based upon the minimum principle of total potential energy. A problem of this type can be found in Jarre et al. [20]; see also Ben-Tal and Bendsøe [6]. Optimization with the continuous design variables often results in a truss design which is not practical in several respects. The member cross-sectional areas of the optimal design can possibly take all different values, which is not acceptable for commercial and manufacturability reasons. Also, existence of too thin members is not accepted. Therefore, formulations and algorithms for truss optimization with discrete design variables have been studied extensively; see, e.g., [25] for a survey on early contributions. One approach is to make use of the continuous optimal solution as a basis for a discrete design. A simple method is rounding each cross-sectional area of the continuous optimal solution to a near discrete predetermined candidate value. Several rounding strategies have been proposed in, e.g., [16, 17, 31, 32]. In general optimality and/or feasibility are not guaranteed by such a heuristic method. Another approach is to directly deal with a discrete optimization problem based upon mixed-integer programming (MIP) methodology. Particularly, mixed-integer nonlinear programming (MINLP) formulations have been studied extensively. Early contributions include the algorithms in [12, 28], which solve a sequence of subproblems that are formulated as mixed-integer linear programming (MILP) problems. Schmit and Fleury [32] proposed a dual method to deal with a separable approximation of the original MINLP problem. The methods [12, 28, 32] cited above do not guarantee global optimality. For truss optimization problems with discrete cross-sectional areas, MINLP approaches with the guaranteed global optimality have been developed in [1, 13, 34]. Also, Stolpe and Kawamoto [35] proposed an MINLP approach considering geometrical nonlinearity in order to solve a design problem of link mechanisms. An outer approximation method for MINLP has been proposed in [23] for solving a certain class of topology optimization problems. An MILP formulation for truss topology optimization under the stress constraints is due to Rasmussen and Stolpe [29]. This formulation, briefly recalled in section 2.3 of this paper, follows naturally from the MILP formulation developed in [36] for continuum-based topology optimization under the stress constraints. Bollapragada et al. [11] proposed a mixed-logical linear programming approach for truss optimization under the displacement constraints. This approach uses a small positive lower bound for the member cross-sectional areas and considers the elongation constraints for the all members included in the ground structure. Therefore, truss topology is not optimized in a strict sense. It should be clear that, in the literature [1, 11 13, 16, 17, 28, 29, 31, 32, 34] cited above, the set of discrete values available for design variables is predetermined. 2

5 MINLP has also been applied to truss optimization with continuous member cross-sectional areas. The methods in [27, 30], however, do not guarantee global optimality. If a number of available discrete sizes and/or a number of members in a ground structure are large, then it is usual that global optimization methods based upon enumeration of solutions consume a lot of time. On the other hand, if the available discrete sizes are closely spaced, it sometimes happens that a solution found by a method without guaranteed global optimality, e.g., rounding the continuous optimal solution, is near optimal. However, the obtained discrete design may use a large number of different available sizes. As pointed out by Templeman [37], such a design has several disadvantages from a practical point of view and a truss design that uses only a small number of different sizes would be much more practical. If a quite small number of available discrete sizes is postulated, then the optimal solution could be expected to use only a limited number of different sizes. For such a problem, rounding the continuous optimal solution is often far less efficient [37] and a global optimization approach could be applicable, provided that the number of design variables is moderately small. However, the optimal solution highly depends on the set of predetermined available discrete sizes; this dependence is concretely shown through the numerical experiments presented in section 5.1. Therefore, it is not clear how one can choose values of widelyspaced sizes, in advance, to assure high quality of mechanical performance of the discrete optimal solution. This paper develops a mixed-integer second-order cone programming (MISOCP) approach that attempts to solve these issues. Namely, to find a truss design that has high performance as far as possible and uses only a small number of different sizes, we directly address an upper bound constraint for the number of different cross-sectional areas. The member cross-sectional areas can be considered either continuous design variables or to be selected among (closely-spaced, if necessary) predetermined available sizes. MISOCP 1 is a class of optimization problems in the form of min f x + r y (1a) s. t. A i x + G i y + b i d i x + e i y + h i, i = 1,..., k, (1b) x {0, 1} n, y R p. (1c) (1d) Here, x and y are variables to be optimized, A i R mi n and G i R mi p (i = 1,..., k) are constant matrices, f R n, r R p, b i R m i, d i R n, and e i R p (i = 1,..., k) are constant vectors, h i R (i = 1,..., k) are constant scalars, and A i x + G i y + b i denotes the standard Euclidean norm of A i x + G i y + b i, i.e., v = (v v) 1/2 for vector v. If we replace binary constraints, (1c), with linear constraints, 0 x j 1 (j = 1,..., n), then the resulting relaxation problem is SOCP. By virtue of this property, MISOCP (1) can be solved globally by using, e.g., a branch-and-bound method; see, e.g., [5, 14, 38] for more account. For solving MISOCP globally, several well-developed software packages, e.g., CPLEX [19] and Gurobi Optimizer [18], are available. In the field of structural optimization, MISOCP has recently been applied to a damper placement optimization problem for supplemental damping of a building structure [22]. The MISOCP formulation proposed in this paper is based upon the SOCP formulation for the continuous compliance optimization of trusses 1 Also called mixed-integer conic quadratic programming. 3

6 mentioned above. Indeed, the former can be viewed as a natural extension of the latter in the sense that the continuous relaxation of the former coincides with the latter. Recently, Lavan and Amir [24] have proposed an MINLP formulation for a closely related structural optimization problem. They considered an optimization problem of the damping coefficients of viscous dampers that are used for seismic retrofitting of a building and addressed the upper bound constraint for the number of different damper sizes. Let x e denote the damping coefficient of damper e and suppose that the upper bound is two. By making use of binary variables s e1 and s e2, this constraint can be written as x e = s e1 [(1 s e2 )y 1 + s e2 y 2 ], e, (2a) s e1, s e2 {0, 1} e, (2b) where y 1, y 2 R are additional variables representing the adopted values of the damping coefficients. In the problem setting considered in this paper, x e is regarded as the cross-sectional area of truss member e. However, due to nonconvexity of constraint (2a), it seems to be difficult to develop an algorithm with guaranteed global optimality; formulation (2) yields an MINLP problem, the continuous relaxation of which is a nonconvex optimization problem. Formulation (2) is originated from the one for multi-phase material topology optimization of continua [15, 33]. By using the idea there, the formulation can be extended to any value of the upper bound; for instance, if the upper bound for different design variables is three, then we obtain x e = s e1 [(1 s e2 )y 1 + s e2 (1 s e3 )y 2 + s e2 s e3 y 3 ], s e1, s e2, s e3 {0, 1}, e, e. This case includes a higher-order term, s e1 s e2 s e3 y 3, than (2). Thus difficulty of the optimization problem depends on the upper bound for different design variables. In contrast, with the formulation developed in this paper, the compliance optimization problem can be formulated as MISOCP irrespective of the upper bound. Again, a remarkable advantage of the MISOCP approach is guaranteed convergence to a global optimal solution and no need for developing algorithms specialized for specific optimization problems. The paper is organized as follows. Section 2 recalls the two existing formulations, the SOCP formulation of the compliance optimization with continuous design variables and the MILP formulation of that with discrete design variables. Section 3 presents the MISOCP formulation for the compliance optimization with the upper bound constraint on the number of different member crosssectional areas. Section 4 explores some practical constraints on the member cross-sectional areas that can be considered within the framework of MISOCP. Section 5 presents numerical experiments. Conclusions are drawn in section 6. A few words regarding notation. All vectors are assumed to be column vectors. The (m + n)- dimensional column vector (z, x ) consisting of z R m and x R n is often written simply as (z, x). For a finite set S, we use S to denote the cardinality of S, i.e., the number of (different) elements in S. For instance, if s 1 = s 2 = 1, s 3 = 3, and s 4 = s 5 = 4, then {s i i = 1,..., 5} = 3. For a, b R with a < b, we denote by [a, b] and ]a, b[ the closed and open intervals between a and 4

7 b, respectively, i.e., [a, b] = {x R a x b}, ]a, b[ = {x R a < x < b}. For x R, we use x to denote the largest integer that is not greater than x. 2 Review of exiting formulations for compliance optimization This section summarizes fundamentals of the compliance optimization of truss structures. Section 2.1 defines this optimization problem. Section 2.2 recalls its SOCP formulation, which serves as a basis for the MISOCP formulations developed in sections 3 and 4. As another relevant problem, section 2.3 recalls the MILP formulation for the problem in which each cross-sectional area is chosen among finitely many predetermined available values. 2.1 Compliance minimization problem Following the conventional ground structure approach, consider a truss consisting of many candidate members connected by nodes. Throughout the paper we assume small deformation and linear elasticity. Let m and d denote the number of members and the number of degrees of freedom of the displacements, respectively. We use x e to denote the cross-sectional area of member e and write x = (x 1,..., x m ) which is the vector of design variables to be optimized. Let u R d and c e R denote the vector of displacements and the elongation of member e, respectively. The compatibility relation between u and c e is written as c e = b e u, e = 1,..., m, (3) where b e R d is a constant vector. The stiffness matrix, denoted K(x) R d d, is given by K(x) = m e=1 Ex e l e b e b e, (4) where l e is the undeformed length of member e and E is Young s modulus. Let f R d denote the vector of external forces. The compliance, denoted π(x) R {+ }, is then defined by π(x) = sup{2f u u K(x)u u R d }. (5) Let V and x max denote the specified upper bounds for the structural volume and for the member cross-sectional area, respectively. The compliance minimization problem is formulated as min π(x) (6a) s. t. m l e x e V, (6b) e=1 0 x e x max, e = 1,..., m. (6c) 5

8 2.2 SOCP formulation for continuous design variables It is known that problem (6) can be recast as an SOCP problem; see, e.g., Makrodimopoulos et al. [26]. The SOCP problem is formulated in variables x e, w e, and q e (e = 1,..., m) as min 2 m e=1 w e (7a) s. t. [ ] w e x e w e + x e, 2le /Eq e e = 1,..., m, (7b) m q e b e = f, (7c) e=1 m l e x e V, e=1 (7d) 0 x e x max, e = 1,..., m. (7e) Here, for each e = 1,..., m, the inequality constraint in (7b) is a second-order cone constraint in terms of w e + x e, w e x e, and 2l e /Eq e. Variables w e and q e correspond to the complementary strain energy and the axial force, respectively, of member e. Details of derivation of problem (7) appear in appendix A. 2.3 MILP formulation for discrete design variables We next suppose that the cross-sectional area of each member is to be chosen among some predetermined candidate values. That is, we add the constraint x e {0, ξ 1,..., ξ p }, e = 1,..., m. (8) to problem (6), where ξ 1,..., ξ p are predetermined positive constants. It is known that this optimization problem can be reformulated as an MILP problem; see e.g., [29]. For member e, we introduce binary variables, τ e1,..., τ ep, to indicate the adopted candidate value for x e, in a manner such that τ ei = 1 means x e = ξ i. Constraint (8) can be rewritten as x e = p ξ i τ ei, i=1 p τ ei 1, i=1 τ ei {0, 1}, i = 1,..., p. By using this expression, the optimization problem can be reformulated as the following MILP 6

9 problem: min s. t. f u m p e=1 i=1 (9a) E ξ i l e χ ei b e = f, (9b) χ ei b e u M(1 τ ei ), e = 1,..., m; i = 1,..., p, (9c) χ ei Mτ ei, e = 1,..., m; i = 1,..., p, (9d) m ( p ) l e ξ i τ ei V, (9e) e=1 i=1 p τ ei 1, e = 1,..., m, (9f) i=1 τ ei {0, 1}, e = 1,..., m; i = 1,..., p. (9g) Here, M 0 is a sufficiently large constant. Variables to be optimized are u, χ ei, and τ ei (e = 1,..., m; i = 1,..., p). 3 Standardization constraints This section presents MISOCP formulations for truss topology optimization with the upper bound constraint for the number of different cross-sectional areas. Section 3.1 deals with a simple case in which a truss should consist of members with uniform cross-sectional areas. This result is generalized in section 3.2 for the constraint that n different cross-sectional areas can be used in a truss design. Section 3.3 shows a scale-free property of the optimization problems in sections 3.1 and Uniformity constraint on member cross-sections In this section we attempt to find a truss design that minimizes the compliance when the truss consists of members that have the same cross-sectional areas. The optimal value of the unified cross-sectional area is explored from a set [0, x max ], where x max is a specified upper bound. This topology optimization problem essentially consists of the following two factors: Determine whether each member has a positive cross-sectional area or vanishes; Determine the common value of cross-sectional areas for existing members. The formulation developed below make use of m binary design variables for the former decision and one continuous design variable for the latter decision. The condition under consideration is satisfied if and only if there exists y [0, x max ] such that x 1,..., x m satisfy x e {0, y}. (10) We rewrite (10) with making use of binary variables t e {0, 1} (e = 1,..., m) that indicate existing members. Namely, t e = 0 means that member e vanishes, i.e., x e = 0, while t e = 1 means member 7

10 e exists, i.e., x e > 0. Then condition (10) can be expressed as x e = { y if te = 1, 0 if t e = 0. (11) In conjunction with x e [0, x max ] and y [0, x max ], we can restate (11) as 0 x e x max t e, x e y x max (1 t e ), which are linear inequality constraints in terms of x e, y, and t e. The observation above can be stated formally as follows. Proposition 3.1. Suppose 0 y x max. Then x e satisfies x e {0, y} (12) if and only if there exist t e satisfying 0 x e x max t e, (13a) x e y x max (1 t e ), (13b) t e {0, 1}. (13c) Proof. Suppose t e = 1 in (13c). Then (13a) and (13b) are reduced to 0 x e x max, x e y 0, which imply x e = y and hence (12) is satisfied. Conversely, if x e = y, then t e = 1 satisfies (13). Alternatively, suppose t e = 0 in (13c). Then (13a) and (13b) are reduced to 0 x e 0, x e y x max, which are equivalent to x e = 0 and y x max and hence (12) is satisfied. Conversely, if x e = 0, then t e = 0 satisfies (13). Remark 3.2. Since too thin members are not allowed to present from a practical point of view, the specified lower bound for the cross-sectional areas of existing members, i.e., the slenderness constraint, are often considered. Let x min (> 0) denote the specified lower bound. With the slenderness constraint the member cross-sectional area is now required to satisfy x e {0} [x min, x max ]. The result in Proposition 3.1 can be extended to this case by replacing (13a) with x min t e x e x max t e for each e = 1,..., m. 8

11 We are now in position to present an MISOCP formulation for truss optimization with uniform member cross-sectional areas. Recall that the conventional compliance optimization problem has been formulated as SOCP problem (7). Application of Proposition 3.1 yields the following problem: min 2 m e=1 w e (14a) s. t. [ ] w e x e w e + x e, 2le /Eq e e = 1,..., m, (14b) m q e b e = f, (14c) e=1 m l e x e V, e=1 (14d) 0 x e x max t e, e = 1,..., m, (14e) x e y x max (1 t e ), e = 1,..., m, (14f) t e {0, 1}, e = 1,..., m. (14g) In this problem, x e, w e, q e (e = 1,..., m) and y are continuous design variables and t e (e = 1,..., m) are binary design variables. At the optimal solution, y becomes equal to the unified value of the cross-sectional areas of the existing members. Inequalities in (14b) are second-order cone constraints. Constraints (14c), (14d), and (14e) are linear constrains. Inequalities in (14f) can be treated as second-order cone constraints in the two-dimensional space or, alternatively, can be reduced to linear inequality constraints as x max (1 t e ) x e y x max (1 t e ). Thus all the constraints other than (14g) are in the forms of linear or second-order cone constraints. Also, the objective function is a linear function. Therefore, problem (14) is an MISOCP problem. Remark 3.3. Problem (7) corresponds to a continuous relaxation of problem (14). To see this, we show that for any x 1,..., x m [0, x max ] there exist t 1,..., t m [0, 1] and y satisfying (14e) and (14f). Indeed, t e = x e /x max (e = 1,..., m) and y = x max satisfy these conditions. 3.2 Upper bound constraint on number of different cross-sections Suppose that we adopt, not necessarily unique, but a limited number of different values for the design variables. Let n denote the specified upper bound for the number of different cross-sectional areas. Then the constraint restricting the variation of cross-sectional areas can be written as {x e e = 1,..., m} \ {0} n. (15) It is worth noting that (15) with n = 1 corresponds to the constraint studied in section 3.1. Like the discussion in section 3.1, we formulate this condition as a system of linear inequalities by making use of some 0-1 variables. The key to this reformulation is that the condition under consideration is satisfied if and only if there exist y 1,..., y n [0, x max ] such that x 1,..., x m satisfy x e {0, y 1,..., y n }. (16) 9

12 For member e, we introduce binary variables s e1,..., s en {0, 1} to represent the relations between x e and y 1,..., y n. Let s ej = 1 imply that cross-sectional area y j is used for member e, i.e., x e = y j s ej = 1. (17) Also, s e1 = = s en = 0 implies that member e is removed, i.e., x e = 0 s e1 = = s en = 0. (18) Without loss of generality we can assume that at most one of s e1,..., s en becomes 1, i.e., n s ej 1 j=1 Then, in conjunction with x e [0, x max ] and y j [0, x max ], condition (17) can be rewritten as x e y j x max (1 s ej ), while condition (18) can be rewritten as n 0 x e x max s ej. The reformulation sketched above can be stated formally as follows. Proposition 3.4. Suppose 0 y j x max (j = 1,..., n). Then x e satisfies x e {0, y 1,..., y n }, (19) if and only if there exist t e and s e1,..., s en satisfying 0 x e x max t e, (20a) x e y j x max (1 s ej ), j = 1,..., n, (20b) n t e = s ej, (20c) j=1 t e 1, (20d) s ej {0, 1}, j = 1,..., n. (20e) Proof. We begin with the only if part, i.e., suppose that (19) holds. If x e > 0, there exists j 1 such that x e = y j1. Let t e = 1, s ej1 = 1, and s ej = 0 ( j j 1 ) to see that (20) is satisfied. Alternatively, if x e = 0, then t e = 0 and s e1 = = s en = 0 satisfy (20). Conversely, suppose that (20) is satisfied. Observe that (20c), (20d), and (20e) imply t e {0, 1}. If t e = 0, then (20a) implies x e = 0. Moreover, (20c) and (20e) imply s e1 = = s en = 0. Therefore, (20b) is reduced to y j x max, which is a redundant constraint. Alternatively, if t e = 1, then (20c) and (20e) imply that there exists j 2 such that s ej2 = 1 and s ej = 0 for all j j 2. Then (20b) is reduced to x e y j2 0, (21) x e y j x max, j j 2. (22) Here, (21) is equivalent to x e = y j2, which satisfies (22). Moreover, (20a) is reduced to 0 x e x max, which is a redundant constraint. Thus (19) is satisfied. 10 j=1

13 In (20) of Proposition 3.4, we can assume that y 1,..., y n satisfy x max y 1 y n 0 without loss of generality. Accordingly, we see that the design optimization problem can be formulated by adding the following constraints to problem (7): 0 x e x max t e, e = 1,..., m, (23a) x max y 1 y n 0, (23b) x e y j x max (1 s ej ), e = 1,..., m; j = 1,..., n, (23c) n t e = s ej, e = 1,..., m, (23d) j=1 t e 1, e = 1,..., m, (23e) s ej {0, 1}, e = 1,..., m; j = 1,..., n. (23f) In (23), all the constraints other than (23f) are linear constraints. As a result, we obtain an MISOCP problem, where the continuous variables are x e, w e, q e (e = 1,..., m), and y 1,..., y n and binary variables are t e and s e1,..., s en (e = 1,..., m). Remark 3.5. Problem (7) serves as a continuous relaxation of the MISOCP problem formulated above. To see this, we show that for any x e [0, x max ] (e = 1,..., m) there exist t e, s ej [0, 1] and y j (e = 1,..., m; j = 1,..., n) satisfying (23a) (23e). For instance, let t e = s e1 = x e /x max, s e2 = = s en = 0 (e = 1,..., m), and y 1 = = y n = x max to see that these constraints are satisfied. Remark 3.6. In (23), constraints (23d), (23e), and (23f) imply that t e can become either 0 or 1. Therefore, we can replace (23e) by t e {0, 1}, e = 1,..., m without changing the feasible set. In the numerical experiments presented in section 5, t 1,..., t m are treated as 0-1 variables. Remark 3.7. Variable t e in (23) serves as a indicator of presence of member e. By using this variable, various constraints on truss topology can be handled within the framework of MISOCP. Two examples are forbiddance of the presence of mutually crossing members and restriction of the number of members connected to a node. These constraints were studied also in, e.g., [13, 27, 34] within the framework of mixed-integer programming. Suppose that members e and e cross each other in the ground structure. Since t e = 1 if and only if member e exists, forbiddance of crossing means that that t e and t e cannot become one simultaneously. This condition can be written as t e + t e 1. (24) This constraint is considered in the numerical examples presented in section 5.3. Concerning the degree of a node, i.e., the number of existing members connected to a node, the upper and lower bound constraints are formulated as follows. Let ρ max denote the specified upper bound. For node 11

14 v, we use E(v) to denote the set of members, in the ground structure, that are connected to node v. Then the upper bound constraint can be written as t e ρ max. e E(v) The lower bound constraint requires to use an additional 0-1 variable that indicates whether node v exists or vanishes; see [13, 34] for details. Remark 3.8. Condition (20) in Proposition 3.4 (and constraint (23) also) allows y j = 0 and/or y j = y j become feasible. It can be also the case that none of x 1,..., x m take the value of y j > 0. In such cases, the number of different cross-sectional areas used in the corresponding truss design is less than n. Consider an example with m = 3 and n = 2. We begin with the case in which the number of different cross-sectional areas is exactly n = 2. Suppose the member cross-sectional areas are x 1 = 10, x 2 = 6, x 3 = 6. Then y j and s ej satisfying (20) are determined uniquely as y 1 = 10, s 11 = 1, s 21 = 0, s 31 = 0, y 2 = 6, s 12 = 0, s 22 = 1, s 32 = 1. Next, suppose that the member cross-sectional areas are x 1 = 10, x 2 = 10, x 3 = 0, i.e., the number of different cross-sectional areas is one (< n). We can see that there exist y j and s ej satisfying (23). Such y j and s ej are not determined uniquely. First, when y 1 = y 2, we see that [ ] [ ] [ ] {[ ] [ ]} [ ] {[ ] [ ]} [ ] [ ] y 1 10 s s s 31 0 =,,,,, = y 2 s 12 satisfy (23). Here, there is no distinguish between adopting y 1 and adopting y 2 for members e = 1 and 2. Second, when 0 < y 2 < y 1, (23) is satisfied with [ ] [ ] [ ] [ ] [ ] [ ] s 11 1 s 21 1 s 31 0 y 1 = 10, y 2 ]0, 10[, =, =, = s 12 In this case, y 2 is not used for any members. Finally, when y 2 = 0, we see that [ ] [ ] [ ] [ ] [ ] [ ] [ ] {[ ] [ ]} y 1 10 s 11 1 s 21 1 s =, =, =,, y 2 s 12 s 22 s 22 s 22 s 32 s 32 s 32 also satisfy (23). Here, s 32 = 1 means that y 2 = 0 is used for member 3. 12

15 3.3 Scale-free property This section establishes a scale-free property of optimal solutions of the problems proposed in section 3.1 and section 3.2. The optimization problem studied in section 3.1 is obtained by adding the constraint x e {0, y}, e = 1,..., m (25) to the conventional compliance minimization problem, (6). Due to this constraint, any feasible solution has uniform cross-sectional areas. On the other hand, a truss design with uniform crosssectional areas can be obtained by predetermining a cross-sectional area for existing members, like the problems reviewed in section 2.3. This corresponds to solving the conventional problem (6) with adding the constraint x e {0, ξ 1 }, e = 1,..., m, (26) where ξ 1 is a predetermined positive constant. However, imposing constraint (25) is essentially different from imposing (26). That is, the optimal solution under constraint (25) has a scale-free property as shown below, while this is not the case for the optimal solution under (26). We recall the scale-free property of the conventional compliance minimization problem, (6). Let x denote the optimal solution of problem (6). Assume that the upper bound constraints for the cross-sectional areas, x e x max, e = 1,..., m, are inactive at x. Let α be a constant satisfying α ]0, x max / max{x 1,..., x m}]. (27) It is worth noting that (27) implies αx e x max (e = 1,..., m). From the linearity of the stiffness matrix on x, (4), and the definition of the compliance, (5), we obtain π(αx ) = 1 α π(x ). (28) Suppose that V in problem (6) is replaced by α V. Then we see that αx is optimal for the problem after this scaling. Alternatively, if l 1,..., l m are replaced by αl 1,..., αl m, then x /α becomes optimal. Therefore, the optimal solution needs only to be computed for one value of V and one geometric scale. The optimal solution corresponding to different values of these quantities can be obtained by applying the scaling explained above. This scale-free property of the conventional compliance optimization of trusses is well known; see, e.g., [8]. In contrast, the optimization problem with constraint (26) does not posses such a scale-free property. This is because, if x satisfies (26), αx with α 1 does not satisfy (26). Similarly, even if we predetermine multiple candidates as x e {0, ξ 1,..., ξ p }, e = 1,..., m, (29) αx does not satisfy (29) in general and the problem lacks the scale-free property. For this reason, when we impose constraint (26) or (29), the optimal truss topology depends on the values of V and 13

16 ξ 1,..., ξ p. Concrete examples of this dependence appear in the numerical experiments presented in section 5.1. The scale-free property of the optimization problem with constraint (25) can be shown as follows. Let (x, y ) be an optimal solution. Then the volume constraint is active at x. Indeed, if it is inactive, i.e., if holds, then we see that (ˇx, ˇy) defined by ˇx = m l e x e < V e=1 V m e=1 l ex e x, ˇy = V m e=1 l ex e y (30) is feasible for the same problem and that (28) yields π(ˇx) = m e=1 l ex e π(x V ) < π(x ). Let α be a constant satisfying (27). Consider an optimization problem obtained by replacing V with α V. Then (αx, αy ) is feasible for this problem; the structural volume of αx is equal to α V. Suppose for contradiction that it is not optimal, i.e., that there exists a feasible solution, denoted (ˇx, ˇy), satisfying π(ˇx) < π(αx ) = π(x )/α. This implies π(ˇx/α) = απ(ˇx) < π(x ). (31) Since (ˇx/α, ˇy/α) is feasible for the original problem with upper bound V, (31) means that (x, y ) is not optimal for the problem with V, which completes the proof. Similar consideration can be made for the case in which l 1,..., l m are replaced by αl 1,..., αl m. Also, the optimization problem proposed in section 3.2 has the same scale-free property. 4 More constraints associated with standardization This section explores three practical constraints on the cross-sectional areas that can be considered together with the standardization constraint introduced in section 3. These constraints can be handled within the framework of MISOCP. 4.1 Exploring optimal unit amount Suppose, for simplicity, that a truss consists of members with only three different cross-sectional areas, denoted y 1, y 2, and y 3. Such a truss can actually be obtained as the optimal solution of the MISOCP problem presented in section 3. From a practical point of view, however, not only the number of different cross-sectional ares but also the values of y j could probably come into an issue. Suppose, for instance, y 3 0. It is usual that very thin members are not accepted in practice. 14

17 Therefore, one may need to round y 3 toward zero and explore a truss design with only two different cross-sectional areas. Alternatively, a practically acceptable lower bound would be given for y 3 and a truss design with three different cross-sectional areas could be explored. On the other hand, suppose that y 3 is practically large but y 1 y 2. In this case, one might be able to explore a truss design with two different cross-sections, probably an intermediate value of y 1 and y 2 and a value close to y 3. In short, when we use some different cross-sectional areas, it might be desired that the values of used cross-sectional areas are distributed almost evenly. This section investigates such a constraint. As a remedy, we may determine the member cross-sectional areas such that there exists υ (> 0) satisfying x e {0, υ, 2υ, 3υ}, e = 1,..., m. Then the cross-sectional areas used in the obtained truss design are, in a sense, equally distributed. It should be clear that υ is considered a design variable. Therefore, the optimization problem retains the scale-free property discussed in section 3.3. In general, letting n denote the upper bound for the number of different cross-sectional areas, we consider the following constraints: x e {0, υ, 2υ,..., nυ}, e = 1,..., m. (32) Here, υ can be interpreted as a unit value of cross-sectional area. In the optimization problem, we explore the optimal unit value as well as the amount of area used for each member. It is worth noting that all of υ, 2υ,..., nυ are not necessarily used in the optimal design. Constraint (32) can be treated with the formulation proposed in section 3.2. Namely, constraint (32) is satisfied if, in (23), y 1,..., y n 1 are multiples of y n. Thus the optimization problem under consideration can be formulated by adding the linear equality constraints, y 1 /n = y 2 /(n 1) = = y n 1 /2 = y n, to the MISOCP problem formulated in section Selection from predetermined finitely many candidates In this section we suppose that the cross-sectional area should be chosen from a set of finitely many predetermined candidates. Due to manufacturing and commercial convenience, it is actually often in practice that a member cross-section is chosen among available candidates. Let ξ i > 0 (i = 1,..., p) denote the predetermined cross-sectional areas, where p is the number of the available candidates. Then the member cross-sectional area is determined as x e {0, ξ 1,..., ξ p }. (33) This constraint can be handled by making use of the concept of multiple-choice constraints in integer programming. In the following we combine this constraint with the upper bound constraint for the number of actually adopted candidates. 15

18 In the same manner as [29, 34], we use 0-1 variables, τ e1,..., τ ep, to represent the cross-sectional area used for member e. Let τ ei = 1 if x e = ξ i, otherwise τ ei = 0. Then constraint (33) can be rewritten as p x e = ξ i τ ei, (34) i=1 p τ ei 1, (35) i=1 τ ei {0, 1}, i = 1,..., p. (36) Note that x e = 0 if and only if τ e1 = = τ ep = 0. The constraint on the variation of cross-sectional areas can be formulated as follows. First, suppose that the truss should consist of members with uniform cross-sectional areas. This means that there exists y R such that x 1,..., x m satisfy x e {0, y} and (33). Like section 3.1, we use variable t e such that t e = 1 means x e > 0 and t e = 0 means x e = 0. From (34), (35), and (36), t e can be represented in terms of τ e1,..., τ ep as p t e = τ ei. (37) i=1 It follows from Proposition 3.1 that the uniformity constraint is satisfied if and only if all the members satisfy x e y x max (1 t e ), (38) where x max = max{ ξ 1,..., ξ p }. The upshot is that the optimization problem under consideration can be formulated as an MISOCP by adding constraints (34), (35), (36), (37), and (38) to problem (7). We next consider the case in which at most n different candidates are allowed to be used as member cross-sectional areas. In the same manner as Proposition 3.4, we use continuous variables y 1,..., y n to denote the cross-sectional areas that are actually used in a truss design. For member e, the selected cross-sectional area is represented by using 0-1 variables s e1,..., s en so that s ej = 1 means x e = y j. This condition can be rewritten as x e y j x max (1 s ej ), j = 1,..., n. (39) It should be clear that x e is subjected to constraints (34), (35), and (36), which is equivalent to (33). Hence, s ej = 1 in (39) implies y j {0, ξ 1,..., ξ p }, although y j is treated as a continuous variable. For member e, exactly one of s e1,..., s en should become 1 if x e > 0, while s e1 = = s en = 0 if x e = 0. This condition can be formulated by using τ e1,..., τ ep as n p s ej = τ ei. (40) j=1 i=1 Consequently, the design optimization problem under consideration can be formulated as an MIS- OCP problem by adding constraints (34), (35), (36), (39), (40), and s ej {0, 1}, j = 1,..., n (41) to problem (7). 16

19 Remark 4.1. Like the formulations presented in section 3, the continuous relaxations of the MISOCP problems formulated in section 4 are SOCP problem (7), and hence the continuous compliance optimization problem in (6). This correspondence might be a particular attribute of the MISOCP formulations presented in this paper. For comparison, we may consider an alternative formulation of the problem studied in this section. Indeed, when the set of candidate cross-sectional areas is given, it is possible to recast the optimization problem as an MILP problem, by adding constraints (39), (40), and (41) to MILP problem (9). The obtained MILP problem then involves M, a sufficiently large constant. For this reason, the continuous relaxation of this MILP problem is in general different from the continuous compliance optimization problem in (6); the feasible set of problem (6) is a subset of that of the MILP problem. Thus the proposed MISOCP problem has a tighter continuous relaxation than the MILP formulation. 4.3 Selection from predetermined multiples As a special case of (33) in section 4.2, we next suppose that the cross-sectional area should be chosen among multiples of a predetermined unit value. In this case the number of integer variables can be reduced drastically, which may speed up the solution process. Suppose that the member cross-sectional area is to be determined as x e {0, δ, 2 δ,..., p δ}, (42) where δ > 0 is a predetermined constant. It is known that (42) can be expressed by using only log 2 p + 1 binary variables; see, e.g., [10]. Namely, (42) can be rewritten as x e = δ r 2 l 1 τ el, (43) l=1 τ el {0, 1}, l = 1,..., r, (44) x e x max, (45) where we use r = log 2 p + 1 and x max = p δ for notational simplicity. 2 The binary expansion in (43), (44), and (45) can be combined with the constraint on the number of different cross-sectional areas as follows. We begin with the uniformity constraint. The key idea is again Proposition 3.1. For member e, we use variable t e such that t e = 0 means x e = 0 and t e = 1 means x e > 0. It follows from (43) and (44) that t e can be related to τ e1,..., τ er as t e = { 0 if τe1 = = τ er = 0, 1 otherwise. (46) Since τ e1,..., τ er are 0-1 variables, relation (46) can be reduced to the following linear inequality constraints: r t e τ el, (47) l=1 t e τ el, l = 1,..., r, (48) t e 1. (49) 2 Constraints (43) and (44) imply x e {0, δ, 2 δ,..., (2 r 1) δ}. 17

20 Then x e should take the same value for all e such that t e = 1, which can be rewritten as x e y x max (1 t e ). (50) Thus we attain an MISOCP formulation by adding constraints (43), (44), (45), (47), (48), (49), and (50) to problem (7). Remark 4.2. Constraints (48) and (49) imply t e {0, 1}, because τ el {0, 1} (l = 1,..., r). Therefore, we can replace (49) by t e {0, 1}. Actually we treat t e as a binary design variable in the numerical experiments presented in section 5. We next suppose that at most n different cross-sectional areas can be used in a truss design. According to Proposition 3.4, we formulate this condition with variables y j R and s ej {0, 1} as (20b). Variables s e1,..., s en are related to variable t e, which serves as an indicator of existence of member e, through (20c). Moreover, relation between t e and τ e1,..., τ er is given by (47) and (48). The upshot is that we add the following constraints to problem (7): x e = r 2 l 1 τ el x max, e = 1,..., m, (51a) l=1 x max y 1 y n 0, (51b) x e y j x max (1 s ej ), e = 1,..., m; j = 1,..., n, (51c) n t e = s ej 1, e = 1,..., m, (51d) t e j=1 r τ el, e = 1,..., m, (51e) l=1 t e τ el, e = 1,..., m; l = 1,..., r, (51f) s ej {0, 1}, e = 1,..., m; j = 1,..., n, (51g) τ el {0, 1}, e = 1,..., m; l = 1,..., r. (51h) Here, all the constraints other than (51g) and (51h) are linear constraints. Therefore, the resulting optimization problem is still an MISOCP problem. Remark 4.3. Since (51d) and (51g) imply t e {0, 1}, we can treat t e as a 0-1 variable. This is done in the numerical experiments in section 5. 5 Numerical experiments This section presents numerical experiments on the proposed MISOCP formulations. We examine computational efficiency and study some properties of optimal solutions with standardization constraints on design variables. MISOCP problems were solved by using CPLEX ver [19]. The data of the problems were prepared in the CPLEX LP file format with MATLAB ver Computation was carried out on two 2.66 GHz 6-Core Intel Xeon Westmere processors with 64 GB RAM. The mip strategy miqcpstrat parameter of CPLEX is set to two (solving an LP relaxation of the MISOCP model at each node) and mip cuts all parameter is set to one (generating all types 18

21 4 m 2 m (a) (b) Figure 1: Example (I). (a) The problem setting; and (b) the optimal solution of the continuous optimization problem. of cuts moderately). These parameters were determined by preliminary numerical experiments. The integrality tolerance in CPLEX is set to 0 and the feasibility tolerance in the simplex method is set to The other parameters of CPLEX are set to the default values. Table 1: Computational results of example (I) in cases A and B. The optimal solutions are shown in Figure 2. Areas (mm 2 ) Case n Obj. (J) y 1 y 2 y 3 Time (s) A-1 B A A B B Table 2: Figure 3. Computational results of example (I) in case C. The optimal solutions are shown in Case Cand. areas (mm 2 ) Obj. (J) Vol. (mm 3 ) Time (s) C1 {0, 1000} C2 {0, 1050} C3 {0, 1100} C4 {0, 700, 1400} C5 {0, 400, 800, 1200} C6 {0, 600, 1200, 1800}

22 (a) (b) (c) (d) (e) Figure 2: The optimal solutions of example (I) in cases A and B. The solutions in (a) cases A and B with n = 1 (i.e., A-1 and B-1); (b) case A with n = 2 (A-2); (c) case A with n = 3 (A-3); (d) case B with n = 2 (B-2); and (e) case B with n = 3 (B-3). 5.1 Example (I) Consider the design domain shown in Figure 1(a), where only the nodes of a ground structure are depicted. Any two nodes are connected by a member, but overlapping of members is avoided by removing the longer member when two members overlap. The ground structure has m = 200 members and d = 46 degrees of freedom of displacements. The leftmost and middle nodes in the bottom row are pin-supported. A vertical force of 100 kn is applied at the bottom rightmost node. The specified upper bound for the structural volume is V = mm 3. The upper bound for the member cross-sectional areas is x max = 2000 mm 2. The elastic modulus is 200 GPa. Similar examples have been solved in [1, 34]. The optimal solution of the conventional compliance minimization problem, i.e., the continuous relaxation problem, is shown in Figure 1(b), where the width of each member is proportional to its cross-sectional area. The optimal value is J. The maximum cross-sectional area in this solution is mm 2. Since members aligned in a straight line have the same cross-sectional areas, this solution has 9 different member cross-sectional areas. As for standardization constraints, we consider four cases: 20

23 (a) (b) (c) (d) (e) (f) Figure 3: The optimal solutions of example (I) in case C. The solutions in (a) case C1; (b) case C2; (c) case C3; (d) case C4; (e) case C5; and (f) case C6. Case A: The adopted values of cross-sectional areas are explored freely. problems formulated in sections 3.1 and 3.2 are solved. The optimization Case B: The adopted values of cross-sectional areas forms a set of multiples of a unit value, which is also a design variable. The optimization problem formulated in section 4.1 is solved. Case C: The cross-sectional areas are selected among a few of predetermined candidates. The optimization problem formulated in section 4.3 without the constraint on the number of different cross-sectional areas is solved. Case D: The cross-sectional areas are selected among a large number of predetermined candidates. The optimization problem formulated in section 4.3 with the constraint on the number of different cross-sectional areas is solved. Note that the volume constraint becomes active at the optimal solutions in cases A and B. With n = 1, i.e., with the uniformity constraint on the cross-sectional areas, the optimal solutions in cases A and B certainly coincide. This optimal solution is shown in Figure 2(a). The member cross-sectional areas and the compliance, together with the computational time required by CPLEX, are reported in Table 1. It is worth noting that, as seen in Figures 1(b) and 2(a), the optimal solution 21

24 with uniform cross-sectional areas has the same topology as the optimal solution of the continuous relaxation problem. The increase of the objective value is about 7.5%. Thus, at the expense of small increase of compliance, we can enjoy the advantage of member standardization. The optimal solutions in case A with n = 2 (called case A-2) and n = 3 (called case A-3) are shown in Figure 2(b) and Figure 2(c), respectively. The computational results are listed in Table 1. As expected, the optimal value decreases as n increases. The set of members used in case A-2 (shown in Figure 2(b)) is different from the one in the continuous relaxation solution (in Figure 1(b)). In contrast, the solution in case A-3 (in Figure 2(c)) has the same members as the continuous relaxation solution. Figures 2(d) and 2(e) show the optimal solutions in case B with n = 2 (called case B-2) and n = 3 (called case B-3). The solution in case B-2 has the same structural topology as that in case A-2. In contrast, the solution in case B-3 is much different from the other solutions. This solution has two arches that are connected to each other by a thin member. In case A-1, the MISOCP problem has m = 200 binary variables. In cases A-3 and B-3, we use nm = 600 variables for s ej and m = 200 variables for t e (the latter is optional as mentioned in Remark 3.6). In case C, we assume that the number of predetermined candidate areas is relatively small. Hence, the upper bound constraint for the number of different cross-sectional areas is not considered. For comparison with cases A and B, we consider six problems: Case C1: x e {0, 1000} in mm 2. Case C2: x e {0, 1050} in mm 2. Case C3: x e {0, 1100} in mm 2. Case C4: x e {0, 700, 1400} in mm 2. Case C5: x e {0, 600, 1200, 1800} in mm 2. Case C6: x e {0, 400, 800, 1200} in mm 2. We use r = 2 in (43) for cases C4, C5, and C6, while r = 1 for cases C1, C2, and C3. optimal solutions are collected in Figure 3. Problems with single candidate area are examined in cases C1, C2, and C3. The candidate values of these problems are determined to be close to the optimal solution in case A-1, i.e., mm 2. The optimal solutions in cases C1 and C3 (shown in Figure 3(a) and Figure 3(c), respectively) include some members different from the members exist in the optimal solution in case A-1 (in Figure 2(a)). In contrast, the optimal solution in case C2 (shown in Figure 3(b)) has the same topology as case A-1. However, the optimal value in case C2 is larger than that in case A-1, because the volume constraint is inactive in case C2. Thus the optimal topology with uniform cross-sectional areas highly depends on the predetermined value of cross-sectional areas. Therefore, it may not be easy to guess the predetermined value such that the optimal solution has the same topology as the solution in case A-1, which has a scale-free property. The optimal solution in case C4 (shown in Figure 3(d)) is very similar to those in cases A-2 and B-2 (in Figure 2(b) and Figure 2(d)). The optimal solution in case C5 has two arches. The optimal solution in case C6 has low rise. The 22